Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Global existence and asymptotic behavior for a mildly degenerate Kirchhoff wave equation with boundary damping

Author: Qiong Zhang
Journal: Quart. Appl. Math. 70 (2012), 253-267
MSC (2010): Primary 35L80, 35B40
DOI: https://doi.org/10.1090/S0033-569X-2012-01281-0
Published electronically: February 3, 2012
MathSciNet review: 2953102
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Abstract | References | Similar Articles | Additional Information

Abstract: In this article a degenerate nonlinear dissipative wave equation of Kirchhoff type with nonlinear boundary damping is considered. We prove the existence, uniqueness and regularity of the global solution of the system when the initial data are small enough and the geometry of the domain satisfies suitable assumptions. We also obtain the polynomial decay property of the global solution.

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  • 1. A. Arosio and S. Garavaldi, On the mildly degenerate Kirchhoff string, Mathematical Methods in Applied Science, 14 (1991), 177-195. MR 1099324 (92c:35072)
  • 2. V. Barbu, I. Lasiecka, and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Transactions of the American Mathematical Society, 357 (2005), 2571-2611. MR 2139519 (2006a:35203)
  • 3. E. H. de Brito, Decay estimates for the generalized damped extensible string and beam equation, Nonlinear Analysis, 8 (1984), 1489-1496. MR 769410 (86k:34056)
  • 4. G.F. Carrier, On the non-linear vibration problem of the elastic string, Quarterly of Applied Mathematics, 3 (1945), 157-165. MR 0012351 (7:13h)
  • 5. M. M. Cavalcanti, U V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, Existence and exponential decay for a Kirchhoff-Carrier model with viscosity, Journal of Mathematical Analysis and Applications, 226 (1998), 40-60. MR 1646453 (99g:35084)
  • 6. P. D'Ancona and S. Spagnolo, Nonlinear perturbations of the Kirchhoff equation, Communications on Pure and Applied Mathematics, 47 (1994), 1005-1029. MR 1283880 (95d:35105)
  • 7. M. Ghisi, Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term, Journal of Differential Equations, 230 (2006), 128-139. MR 2270549 (2008a:35196)
  • 8. R. Izaguirrea, R. Fuentesb and M. M. Miranda, Existence of local solutions of the Kirchhoff-Carrier equation in Banach spaces, Nonlinear Analysis, 68 (2008), 3565-3580. MR 2401368 (2009d:35233)
  • 9. G. Kirchhoff, Vorlesungen $ \ddot {u}$ber Mechanik, Teubner, Leipzig, 1883.
  • 10. V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Masson, Paris, John Wiley and Sons, Chichester, 1994. MR 1359765 (96m:93003)
  • 11. I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometric conditions, Applied Mathematics and Optimization, 25 (1992), 189-224. MR 1142681 (93b:93099)
  • 12. I. Lasiecka and J. Ong, Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation, Communications in Partial Differential Equations, 24 (1999), 2069-2107. MR 1720766 (2001c:35154)
  • 13. J. L. Lions, On some questions in boundary value problem of mathematical physics, in: G.M. de La Penha, L.A. Medeiros (Eds.), Contemporary development in continuum mechanics and partial differential equations, North-Holland, Amsterdam, 1978, 285-346. MR 519648 (82b:35020)
  • 14. K. Nishihara, Global existence and asymptotic behaviour of the solution of some quasilinear hyperbolic equation with linear damping, Funkcialaj Ekvacioj, 32 (1989), 343-355. MR 1040163 (91h:35213)
  • 15. K. Nishihara and Y. Yamada, On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkcialaj Ekvacioj, 33 (1990), 151-159. MR 1065473 (91f:35181)
  • 16. K. Ono, Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, Journal of Differential Equations 137 (1997), 273-301. MR 1456598 (98f:35149)
  • 17. M. L. Santos, J. Ferreira, D. C. Pereira and C. A. Raposo, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlinear Anaysis, 54 (2003), 959-976. MR 1992515 (2004e:35152)
  • 18. J. Simon, Compact sets in the space $ L^p(0,T;B)$, Annali di Matematica pura ed applicata, 146 (1987), 65-96. MR 916688 (89c:46055)
  • 19. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Applied Mathematical Sciences, Vol. 68, New York, 1988. MR 953967 (89m:58056)
  • 20. Q. Zhang, Global solution for a quasi-linear plate system with boundary memory damping, IMA Journal of Applied Mathematics, 2009. MR 2507295 (2010j:35553)

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Additional Information

Qiong Zhang
Affiliation: Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China
Email: zhangqiong@bit.edu.cn, qiongzhg@gmail.com

DOI: https://doi.org/10.1090/S0033-569X-2012-01281-0
Keywords: Degenerate equation, Kirchhoff equation, boundary dissipation, global existence, polynomial decay.
Received by editor(s): May 4, 2010
Published electronically: February 3, 2012
Additional Notes: The work is supported by the NSF of China (60504001, 60974033), SRF for ROCS, SEM, China (20080732041).
Article copyright: © Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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